Randomized Online Graph Coloring (Preliminary Version)

In this paper we study the problem of coloring graphs in an online manner. The only known deterministic online graph coloring algorithm with a sublinear performance function was found by Lov&z, Saks and Trotter [5]. Their algorithm colors graphs of c h r s matic number x with no more than ((2xn)/log*n) colors, where n is the number of vertices. They point out that the performance can be improved slightly for graphs with bounded chromatic number. For 3chromatic graphs the number of colors used, for example, is O( n log log log n/ log log n) . We show that randomization helps in coloring graphs online. We present a simple randomized online algorithm to color graphs with expected number of colors 0 ( 2 X ~ ~ n s ( l o g n ) x 1 ) . For 3-colorable graphs the expected number of colors our algorithm uses is O(d*). All our algorithms run in polynomial time. It is interesting to note that our algorithm compares well with the best known polynomial time ofline algorithms. For instance, the best polynomial time algorithm known for 3-colorable graphs, due to Avrim Blum, uses O(n3/" poly-log(n)) colors [3]. In section 3 we prove a lower bound of a(&( 1 2 ] ~ ~ ~ l ) ) X 1 ) for the randomized model. No lower bound for the randomized model was previously known. Our result improves even the best known lower bound for the deterministic case: a(( ,$:Gn)X-') for bounded x, due to Noga Alon [l]. 1